The Problem of Tail of Probability Distribution

Like most things in life, we must accept that the occurrence probability of any future event cannot be exactly zero. We must be contented with a structural failure probability that is negligible compared to other risks that people willingly take, such as car accidents. It is generally agreed that adequate safety of engineering structures is achieved by specifying a failure probability of 10−6 per lifetime as the maximum admissible in design [Nordic Committee for Building Structures (NKB) 1978; Melchers 1987; Duckett 2005; Ellingwood 2006]. This probability limit is generally accepted for engineering structures, whether bridges or aircraft (Duckett 2005; Department of National Defense of Canada 2007), although for some nuclear plant structures an even smaller limit is required.

The smallness of this probability limit is a source of great difficulty. To check the design merely by an experimental histogram, at least 108 tests of identical structures or specimens would be required. Even a direct computational verification would necessitate about 108 repetitions of Monte Carlo simulations with a fully realistic material model. Therefore, estimations of loads of such a small failure probability must rely on a model that is justified by a sound theory and is validated by experiments other than histogram testing.

For many years, realistic theoretical models have been available for the probability of ductile and brittle failures. The Gaussian and Weibull distributions, respectively, fit this purpose. Failures of structures made of quasibrittle materials are more difficult to predict and have been researched only recently. The difficulty is that the quasibrittle failures are transitional in nature between ductile and plastic failures.

Quasibrittle materials are heterogeneous materials with brittle constituents. At the scale of normal laboratory testing, they include concretes as the archetypical case, fiberpolymer composites, fiber-reinforced concretes, toughened ceramics, many rocks, coal, sea ice, wood, consolidated snow, particle board, rigid foams, particulate nanocomposites, biological shells, mortar, masonry, fiber-reinforced concrete, asphalt concrete (at low temperatures), stiff clays, silts, cemented sands, grouted soils, particle board, various refractories, bone, cartilage, dentine, dental ceramics, paper, carton, and cast iron.